Università Roma Tre

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Convergence of cluster expansion for the high and low temperature expansions for the pressure of the d-dimensional Ising model
- The exact solution of the 2D Ising model with h=0

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html